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Comprehensive List of Mathematical Symbols

Comprehensive List of Mathematical Symbols

Comprehensive List of Mathematical Symbols

Comprehensive List of Mathematical Symbols

Comprehensive List of Mathematical Symbols

For the corresponding web guides, see Mathematical Symbols.

Table of Contents

1 Constant ...... 3 1.1 Key Mathematical ...... 3 1.2 Key Mathematical Sets ...... 4 1.3 Key Mathematical Infinities ...... 5 1.4 Other Key Mathematical Objects ...... 6 2 Variables ...... 6 2.1 Variables for Numbers ...... 6 2.2 Variables in Geometry ...... 7 2.3 Variables in ...... 7 2.4 Variables in Linear ...... 8 2.5 Variables in Theory and Logic ...... 8 2.6 Variables in Probability and Statistics ...... 9 3 Delimiters ...... 10 3.1 Common Delimiters ...... 10 3.2 Other Delimiters ...... 10 4 Operators ...... 11

2 Table of Contents

Comprehensive List of Mathematical Symbols

4.1 Common Operators ...... 11 4.2 -related Operators ...... 12 4.3 Elementary Functions ...... 12 4.4 Algebra-related Operators ...... 13 4.5 Geometry-related Operators ...... 14 4.6 Logic-related Operators ...... 15 4.7 Set-related Operators ...... 16 4.8 Vector-related Operators ...... 16 4.9 -related Operators ...... 17 4.10 Probability-related Operators ...... 18 4.11 Statistics-related Operators ...... 18 4.12 Key Probability Functions and Distributions .. 19 4.13 Calculus-related Operators ...... 20 5 Relational Symbols ...... 21 5.1 Equality-based Relational Symbols ...... 21 5.2 Comparison-based Relational Symbols ..... 21 5.3 -related Relational Symbols ...... 22 5.4 Geometry-related Relational Symbols ..... 22 5.5 Set-related Relational Symbols ...... 22 5.6 Logic-related Relational Symbols ...... 23 5.7 Probability-related Relational Symbols ..... 23 5.8 Calculus-related Relational Symbols ...... 24 6 Notational Symbols ...... 24 6.1 Common Notational Symbols ...... 24 6.2 Notational Symbols in Geometry and Trigonom- etry ...... 25 6.3 Notational Symbols in Calculus ...... 25 6.4 Notational Symbols in Probability and Statistics 26 7 Additional Resources ...... 26

1 Constant

1.1 Key Mathematical Numbers

1.1 Key Mathematical Numbers 3

Comprehensive List of Mathematical Symbols

Symbols LaTeX Code Example (Explanation)

0 $0$ 3 + 0 = 3 (Zero, ) 1 $1$ 5 × 1 = 5 (One, multiplicative identity) √ √ √ 2 $\sqrt{2}$ ( 2 + 1)2 = 3 + 2 2 ( root of 2) e $e$ ln(e2) = 2 (Euler’s constant) 2 π 1 1 ··· π $\$ = 2 + 2 + (Pi, Archimedes’ 6 1 2 constant) √ 1 + 5 φ $\varphi$ φ = (Phi, golden ratio) 2 i $i$ (1 + i)2 = 2i ()

1.2 Key Mathematical Sets

Symbols LaTeX Code Example (Explanation)

∅ $\varnothing$ |∅| = 0 () N $\mathbb{N}$ ∀x, y ∈ N, x + y ∈ N (Set of natural numbers) Z $\mathbb{Z}$ N ⊆ Z (Set of )

4 1.2 Key Mathematical Sets

Comprehensive List of Mathematical Symbols

Z+ $\mathbb{Z}_+$ 3 ∈ Z+ (Set of positive integers) √ Q $\mathbb{Q}$ 2 ∈/ Q (Set of rational numbers) R $\mathbb{R}$ ∀x ∈ R (x2 ≥ 0) (Set of real numbers)

R+ $\mathbb{R}_+$ ∀x, y ∈ R+ (xy ∈ R+) (Set of positive real numbers) C $\mathbb{C}$ ∃z ∈ C (z2 + 1 = 0) (Set of complex numbers)

Zn $\mathbb{Z}_n$ In the world of Z2, (Set of modulo 1 + 1 = 0. n) R3 $\mathbb{R}^3$ (5, 1, 2) ∈ R3 (Three-dimensional )

1.3 Key Mathematical Infinities

Symbols LaTeX Code Example (Explanation)

ℵ0 $\aleph_0$ ℵ0 + 5 = ℵ0 (Cardinality of natural numbers) ℵ c $\mathfrak{c}$ c = 2 0 (Cardinality of real numbers) ω $\omega$ ∀n ∈ N (n < ω) (Smallest infinite ordinal number)

1.3 Key Mathematical Infinities 5

Comprehensive List of Mathematical Symbols

1.4 Other Key Mathematical Objects

Symbols LaTeX Code Example (Explanation)

0 $\mathbf{0}$ ∀v ∈ V, v + 0 = v (Zero vector) e $e$ e ◦ e = e ( of a ) I $I$ AI = IA = A (Identity matrix) Z C $C$ 1 dx = x + C () > $\top$ For each proposition P , (Tautology) P ∧ > ≡ P . ⊥ $\bot$ For each proposition P , (Contradiction) P ∧ ¬P ≡ ⊥. Z $Z$ Z ∼ N(0, 1) (Standard normal distribution)

2 Variables

2.1 Variables for Numbers

Symbols LaTeX Code Example (Explanation)

m, n, p, q $m$, $n$, $p$, $q$ m + n − q = 1 (Integers and natural numbers)

6 2.1 Variables for Numbers

Comprehensive List of Mathematical Symbols

a, b, c $a$, $b$, $c$ ax + by = 0 (Coefficients for functions and equations) x, y, z $x$, $y$, $z$ If 2x + 5 = 3, then (Unknowns in x = −1. functions and equations) ∆ $\Delta$ ∆ = b2 − 4ac for (Discriminant) quadratic X10 i, j, k $i$, $j$, $k$ i = 55 (Index variables) i=1 t $t$ At t = 5, the velocity (Time variable) is v(5) = 32. z $z$ zz¯ = |z|2 (Complex numbers)

2.2 Variables in Geometry

Symbols LaTeX Code Example (Explanation)

P , Q, R, S $P$, $Q$, $R$, $S$ PQ ⊥ QR (Vertices)

ℓ $\ell$ ℓ1 k ℓ2 (Lines) α, β, γ, θ $\alpha$, $\beta$, α + β + θ = 180◦ (Angles) $\gamma$, $\theta$

2.3 Variables in Calculus

Symbols LaTeX Code Example (Explanation)

2.3 Variables in Calculus 7

Comprehensive List of Mathematical Symbols

f(x), g(x, y), h(z) $f(x)$, $g(x,y)$, f(2) = g(3, 1) + 5 (Functions) $h(z)$ 3 an, bn, cn $a_n$, $b_n$, an = () $c_n$ n + 2 eh − e0 h, ∆x $h$, $\Delta x$ lim = 1 → (Limiting variables in h 0 h ) δ, ε $\delta$, For all ε > 0, there is a (Small quantities in $\varepsilon$ δ > 0 such that |x| < δ proofs involving implies that |2x| < ε. limits) F (x), G(x) $F(x)$, $G(x)$ F 0(x) = f(x) ()

2.4 Variables in

Symbols LaTeX Code Example (Explanation)

u, v, w $\mathbf{u}$, 3u + 4v = w (Vectors) $\mathbf{v}$, $\mathbf{w}$ A, B, C $A$, $B$, $C$ AX = B (Matrices) λ $\lambda$ Av = λv (Eigenvalues)

2.5 Variables in Set Theory and Logic

Symbols LaTeX Code Example (Explanation)

A, B, C $A$, $B$, $C$ A ⊆ B ∪ C (Sets)

8 2.5 Variables in Set Theory and Logic

Comprehensive List of Mathematical Symbols

a, b, c $a$, $b$, $c$ a ∈ A (Elements) P , Q, R $P$, $Q$, $R$ P ∨ ¬P ≡ > (Propositions)

2.6 Variables in Probability and Statistics

Symbols LaTeX Code Example (Explanation)

X, Y , Z $X$, $Y$, $Z$ E(X + Y ) = (Random variables) E(X) + E(Y )

µ $\mu$ H0 : µ = 5 (Population means)

σ $\sigma$ σ1 = σ2 (Population standard deviations) s $s$ s =6 σ (Sample standard deviations) n $n$ For n ≥ 30, use the (Sample sizes) normal distribution.

ρ $\rho$ Ha : ρ < 0 (Population correlations) r $r$ If r = 0.75, then (Sample correlations) r2 = 0.5625. π $\pi$ π = 0.5 (Population proportions) X p $p$ p = (Sample proportions) n

2.6 Variables in Probability and Statistics 9

Comprehensive List of Mathematical Symbols

3 Delimiters

3.1 Common Delimiters

Symbols LaTeX Code Example (Explanation)

. $.$ 25.9703 (Decimal separator) : $:$ 1 : 4 : 9 = 3 : 12 : 27 (Ratio indicator) , $,$ (3, 5, 12) (Object separator) (), [], {} $()$, $[]$, $\{ \}$ (a + b) × c (Order-of- indicators) (), [] $()$, $[]$ 3 ∈/ (3, 4], 4 ∈ (3, 4] (Interval indicators)

3.2 Other Delimiters

Symbols LaTeX Code Example (Explanation)       a 1 4 (), [], x y ,   $()$, $[]$,   b $\begin{pmatrix} x 3 6 (Vector/matrix & y \end{pmatrix}$, indicators) $\begin{bmatrix} a \\ b \end{bmatrix}$ {} $\{ \}$ {π, e, i} (Set builder) |, : $\mid, :$ {x ∈ R | x2 − 2 = 0} (“Such that” markers)

10 3.2 Other Delimiters

Comprehensive List of Mathematical Symbols

||, kk $| |, \| \|$ k(3, 4)k = 5 (Metric-related operators)   f(x) x ≥ a 1 x ≥ 0  $\begin{cases} f(x) f(x) =  g(x) x < a & x \ge a \\ g(x) & 0 x < 0 (Piecewise-function x < a \end{cases}$ marker) hi $\langle \rangle$ hka, bi = kha, bi (Inner operator) de $\lceil \rceil$ d2.476e = 3 (Ceiling operator) bc $\lfloor \rfloor$ bπc = 3 (Floor operator)

4 Operators

4.1 Common Operators

Symbols LaTeX Code Example (Explanation) x + y $x+y$ 2a + 3a = 5a (Sum) x − y $x-y$ 11 − 5 = 6 (Difference) −x $-x$ −3 + 3 = 0 () x × y, x · y, xy $x \times y$, (m + 1)n = mn + n (Product) $x \cdot y$, $xy$ x ÷ y, x/y $x \div y$, $x/y$ 152 ÷ 3 = 50.6 (Quotient)

4.1 Common Operators 11

Comprehensive List of Mathematical Symbols

x 53 + 5 53 5 $\displaystyle = + y \frac{x}{y}$ 6 6 6 () xy $x^y$ 34 = 81 (Power) √ −b ± ∆ x ± y $x \pm y$ (Plus and minus) 2a √ √ x $\sqrt{x}$ 2 ≈ 1.414 (Positive ) |x| $|x|$ |x − 3| < 5 (Absolute value) . x x% $x \%$ x% = (Percent) 100

4.2 Function-related Operators

Symbols LaTeX Code Example (Explanation)

dom f $\operatorname{dom}f$ If g(x) = ln x, then (Domain) dom(g) = R. ran f $\operatorname{ran}f$ If h(y) = sin y, then (Range) ran(h) = [−1.1]. f(x) $f(x)$ g(5) = g(4) + 3 (Image of an element) f(X) $f(X)$ f(A∩B) ⊆ f(A)∩f(B) (Image of a set) f ◦ g $f \circ g$ If g(3) = 5 and f(5) = (Composite 8, then (f ◦ g)(3) = 8. function)

4.3 Elementary Functions

12 4.3 Elementary Functions

Comprehensive List of Mathematical Symbols

Symbols LaTeX Code Example (Explanation)

n 0 knx + ··· + k0x $k_n x^n + \cdots The (Polynomial) + k_0x^0$ x3 + 2x2 + 3 has a root in (−3, −2). ex, exp x $e^x$, $\exp x$ ex+y = ex · ey (Natural ) bx $b^x$ 2x > x2 for large x. (General exponential function) ln x $\ln x$ ln(x2) = 2 ln x (Natural logarithmic function) log x $\log x$ log 10000 = 4 (Common logarithmic function) ln x logb x $\log_b x$ log2 x = (General logarithmic ln 2 function) sin x $\sin x$ sin π = 0 ( function) √ π 2 cos x $\cos x$ cos = (Cosine function) 4 2 sin x tan x $\tan x$ tan x = ( function) cos x

4.4 Algebra-related Operators

Symbols LaTeX Code Example (Explanation) gcd(x, y) $\gcd (x,y)$ gcd(35, 14) = 7 (Greatest common factor)

4.4 Algebra-related Operators 13

Comprehensive List of Mathematical Symbols

bxc $\lfloor x \rfloor$ b3.6c = 3 (Floor operator) dxe $\lceil x \rceil$ dπe = 4 (Ceiling operator) min(A) $\min (A)$ If min(A) = 3, then (Minimum) min(A + 5) = 8. max(A) $\max (A)$ max(A ∪ B) ≥ max(A) (Maximum) x mod y $x\bmod y$ 36 mod 5 = 1 (Modulo operator) Xn X5 2 ai $\displaystyle i = 55 i=m \sum_{i=m}^n a_i$ i=1 (Summation) Yn Yn ai $\displaystyle = n! i=m \prod_{i=m}^n a_i$ i=1 (Pi Product) . [a] $[a]$ [a] = {x | xRa} (Equivalence class) deg f $\deg f$ deg(2x2 + 3x + 5) = 2 (Degree of polynomial) z¯ $\bar{z}$ 5 − 8i = 5 + 8i () |z| $|z|$ |eπi| = 1 ( of ) π arg z $\arg z$ arg(1 + i) = + 2πn (Arguments of 4 complex number)

4.5 Geometry-related Operators

Symbols LaTeX Code Example (Explanation)

14 4.5 Geometry-related Operators

Comprehensive List of Mathematical Symbols

∠ABC $\angle ABC$ ∠ABC = ∠CBA (Angle) ∡ABC, m∠ABC $\measuredangle ∡ABC = ∡A0B0C0 (Measure of angle) ABC$, $m\angle ABC$ ←→ ←→ ←→ AB $\overleftrightarrow AB = BA (Infinite line) {AB}$ AB $\overline{AB}$ If B =6 B0, then (Line segment) AB 6= AB0. −→ −→ ∼ −−→ AB $\overrightarrow AB = CD (Ray) {AB}$ |AB| $|AB|$ |AB| < |A0B0| (Distance between two points) ∼ 0 0 0 4ABC $\triangle ABC$ 4ABC = 4A B C (Triangle) □ABCD $\square ABCD$ □ABCD = □DCBA (Quadrilateral)

4.6 Logic-related Operators

Symbols LaTeX Code Example (Explanation)

¬P $\lnot P$ ¬(1 = 2) (Negation) P ∧ Q $P \land Q$ P ∧ Q ≡ Q ∧ P (Conjunction) P ∨ Q $P \lor Q$ πe ∈ Q ∨ πe ∈/ Q (Disjunction) P → Q $P \to Q$ P → Q ≡ (¬P ∨ Q) (Conditional) P ↔ Q $P \leftrightarrow Q$ P ↔ Q =⇒ P → Q (Biconditional)

4.6 Logic-related Operators 15

Comprehensive List of Mathematical Symbols

∀xP (x) $\forall x P(x)$ ∀y ∈ N (y + 1 ∈ N) (Universal statement) ∃xP (x) $\exists x P(x)$ ∃z (z2 = −π) (Existential statement)

4.7 Set-related Operators

Symbols LaTeX Code Example (Explanation)

A, Ac $\overline{A}$, A = A (Complement) $A^{c}$ A ∩ B $A \cap B$ {2, 5} ∩ {1, 3} = ∅ (Intersection) A ∪ B $A \cup B$ N ∪ Z = Z (Union) A/B, A − B $A/B$, $A-B$ In general, (Set difference) A − B =6 B − A. A × B $A \times B$ (11, −35) ∈ N × Z (Cartesian product) P(A) $\mathcal{P}(A)$ P(∅) = {∅} (Power set)

|A| $|A|$ |N| = ℵ0 (Cardinality)

4.8 Vector-related Operators

Symbols LaTeX Code Example (Explanation)

kvk $\| \mathbf{v} \|$ k(3, 4)k = 5 ( of vector)

16 4.8 Vector-related Operators

Comprehensive List of Mathematical Symbols

u · v $\mathbf{u} \cdot u · u = kuk2 (Dot product) \mathbf{v}$ u × v $\mathbf{u} \times u × u = 0 (Cross product) \mathbf{v}$ projv u $\operatorname{proj} proj(0,1)(5, 4) = (0, 4) (Projection vector) _{\mathbf{v}} \mathbf{u}$ span(S) $\operatorname{span} span({i, j}) = R2 (Span of vectors) (S)$ dim(V ) $\dim(V)$ dim(R3) = 3 (Dimension of )

4.9 Matrix-related Operators

Symbols LaTeX Code Example (Explanation)

A + B $A+B$ A + X = B (Matrix sum) A − B $A-B$ In general, (Matrix difference) A − B =6 B − A. −A $-A$ B + (−B) = 0 (Additive inverse) kA $kA$ (−1)A = −A (Scalar product) AB $AB$ AI = IA = A (Matrix product) AT $A^T$ IT = I (Matrix transpose) A−1 $A^{-1}$ (AB)−1 = B−1A−1 (Matrix inverse) tr(A) $\operatorname{tr} tr(AT ) = tr(A) (Trace of matrix) (A)$

4.9 Matrix-related Operators 17

Comprehensive List of Mathematical Symbols

x y 1 4 | | − − det(A), A , $\det(A)$, $|A|$, = 2 12 = 10 w z $\begin{vmatrix} x & 3 2 () y \\ w & z \end{vmatrix}$

4.10 Probability-related Operators

Symbols LaTeX Code Example (Explanation)

n! $n!$ 4! = 4 · 3 · 2 · 1 () nP r $nPr$ 5P 3 = 5 · 4 · 3 ()      n 5 5 nCr,   $nCr$, $\displaystyle   =   r \binom{n}{r}$ 2 3 () P (E) $P(E)$ P (A ∪ B ∪ C) = 0.3 (Probability of event) P (A ∩ B) P (A|B) $P(A|B)$ P (A|B) = (Conditional P (B) probability) E(X) $E(X)$ E(X + Y ) = (Expected value of E(X) + E(Y ) random variable) V (X) $V(X)$ V (5X) = 25V (X) (Variance of random variable)

4.11 Statistics-related Operators

Symbols LaTeX Code Example (Explanation)

18 4.11 Statistics-related Operators

Comprehensive List of Mathematical Symbols

X $\overline{X}$ 3X = 3X (Sample mean) P − 2 2 2 (X X) s $s^2$ s = − (Sample variance) n 1 P (X − µ)2 σ2 $\sigma^2$ σ2 = (Population n variance)

4.12 Key Probability Functions and Distri- butions

Symbols LaTeX Code Example (Explanation)

Bin(n, p) $\operatorname{Bin} If X stands for the (Binomial (n, p)$ number of heads in 10 distribution) coin tosses, then X ∼ Bin(10, 0.5). Geo(p) $\operatorname{Geo} Y ∼ Geo(1/5), then (Geometric (p)$ E(Y ) = 5. distribution) U(a, b) $U(a,b)$ If X ∼ U(3, 7), then (Continuous (7 − 3)2 V (X) = . uniform 12 distribution) N(µ, σ2) $N(\mu, \sigma^2)$ If X ∼ N(3, 52), then (Normal X − 3 ∼ Z. distribution) 5 zα $z_{\alpha}$ z0.05 ≈ 1.645 (Critical z-score) tα,ν $t_{\alpha, \nu}$ t0.05,1000 ≈ z0.05 (Critical t-score) 2 2 ≈ χα,ν $\chi^2_{\alpha, \nu}$ χ0.05,30 43.77 (Critical Chi- squared-score)

4.12 Key Probability Functions and Distributions 19

Comprehensive List of Mathematical Symbols

≈ Fα,ν1,ν2 $F_{\alpha, \nu_1, F0.05,20,20 2.1242 (Critical F-score) \nu_2}$

4.13 Calculus-related Operators

Symbols LaTeX Code Example (Explanation) n + 3 1 lim a $\displaystyle \lim_ lim = n→∞ n n→∞ 2n 2 ( of {n \to \infty} a_n$ ) π sin x lim f(x) $\displaystyle \lim_ lim = x→c → {x \to c} f(x)$ x 3 2 (Limit of function) π lim sin x 2 x→3 sup(A) $\sup(A)$ sup( [−3, 5) ) = 5 (Supremum) ( ) 1 1 inf(A) $\inf(A)$ If B = , ,... , then (Infimum) 1 2 inf(B) = 0. f 0, f 00, f 000, f (n) $f'$, $f''$, $f'''$, (sin x)000 = − cos x () $f^{(n)}$ Z Z b 1 1 π f(x) dx $\displaystyle \int_a^ 2 = a b f(x)\,\mathrm{d}x$ 0 1 + x 4 (Definite ) Z Z f(x) dx $\displaystyle \int f(x) ln x dx = x ln x − x (Indefinite \,\mathrm{d}x$ integral) 2 3 fx $f_x$ If f(x, y) = x y , then 3 () fx(x, y) = 2xy .

20 4.13 Calculus-related Operators

Comprehensive List of Mathematical Symbols

5 Relational Symbols

5.1 Equality-based Relational Symbols

Symbols LaTeX Code Example (Explanation) x = y $x = y$ 3x − x = 2x (Equal) x =6 y $x \ne y$ 2 =6 3 (Non-equal) x ≈ y $x \approx y$ π ≈ 3.1416 (Approximately equal) x ∼ y, xRy $x \sim y$, $xRy$ xRy if and only if (Related to) |x| = |y| x ≡ y $x \equiv y$ 2 ≡ 101 in mod 33 (Equivalent to) f(x) ∝ g(x) $f(x) \propto g(x)$ V ∝ r3 (Proportional to)

5.2 Comparison-based Relational Symbols

Symbols LaTeX Code Example (Explanation) x < y $x < y$ sin x < 3 (Less than) x > y $x > y$ π > e (Greater than) x ≤ y $x \le y$ n! ≤ nn (Less than or equal to)

5.2 Comparison-based Relational Symbols 21

Comprehensive List of Mathematical Symbols

x ≥ y $x \ge y$ x2 ≥ 0 (Greater than or equal to)

5.3 Number-related Relational Symbols

Symbols LaTeX Code Example (Explanation)

m | n $m \mid n$ 101 | 1111 (Divisibility) m ⊥ n $m \perp n$ 31 ⊥ 97 (Coprime integers)

5.4 Geometry-related Relational Symbols

Symbols LaTeX Code Example (Explanation)

ℓ1 k ℓ2 $\ell_1 \parallel PQ k RS (Parallel) \ell_2$ −→ −−→ ℓ1 ⊥ ℓ2 $\ell_1 \perp \ell_2$ AB ⊥ BC (Perpendicular) F ∼ F 0 $F \sim F'$ 4ABC ∼ 4DEF (Similar figures) ∼ 0 ∼ F = F $F \cong F'$ □ABCD = □P QRS (Congruent figures)

5.5 Set-related Relational Symbols

Symbols LaTeX Code Example (Explanation)

22 5.5 Set-related Relational Symbols

Comprehensive List of Mathematical Symbols

2 a ∈ A $a \in A$ ∈ R (Member of) 3 a∈ / A $a \notin A$ π∈ / Q (Not a member of) A ⊆ B $A \subseteq B$ A ∩ B ⊆ A (Subset of) A = B $A = B$ If A = B, then A ⊆ B. (Equal Sets)

5.6 Logic-related Relational Symbols

Symbols LaTeX Code Example (Explanation)

P =⇒ Q $P \implies Q$ x is even =⇒ (Implies) 2 divides x P ⇐= Q $P \impliedby Q$ x = 3 ⇐= 3x + 2 = 11 (Implied by) P ⇐⇒ Q, $P \iff Q$, x =6 y ⇐⇒ P ≡ Q $P \equiv Q$ (x − y)2 > 0 (If and only if) P ∴ Q $P \therefore Q$ i ∈ C ∴ ∃z (z ∈ C) (Therefore) π P ∵ Q $P \because Q$ x = ∵ 2 (Because) sin x = 1 and cos x = 0

5.7 Probability-related Relational Symbols

Symbols LaTeX Code Example (Explanation)

A ⊥ B $A \perp B$ If A ⊥ B, then (Independent P (A ∩ B) = events) P (A) ∩ P (B).

5.7 Probability-related Relational Symbols 23

Comprehensive List of Mathematical Symbols

X ∼ F $X \sim F$ Y ∼ Bin(30, 0.4) (X follows distribution F )

5.8 Calculus-related Relational Symbols

Symbols LaTeX Code Example (Explanation) x f(x) ∼ g(x) $f(x) \sim g(x)$ π(x) ∼ (Asymptotically ln x equal) f(x) ∈ O(g(x)) $f(x) \in O(g(x))$ 2x2 + 3x + 3 ∈ O(x2) (In the big-O of)

6 Notational Symbols

6.1 Common Notational Symbols

Symbols LaTeX Code Example (Explanation)

..., ··· $\ldots$, $\cdots$ 12 + 22 + ··· + n2 (Horizontal )    a11 ··· a1n      . ..  . .. .  ., . $\vdots$, $\ddots$  . . .  (Vertical ellipsis)   am1 ··· amn f : A → B, $f : A \to B$, $A A function g : N → R A →f B \overset{f}{\to} B$ can be thought of as a (Function’s sequence. domain/codomain specifier)

24 6.1 Common Notational Symbols

Comprehensive List of Mathematical Symbols

x 7→ f(x) $x \mapsto f(x)$ The function x 7→ x2 is (Function increasing in the mapping rule) interval [0, ∞). Q.E.D., ■, □ $Q. E. D.$, Thus the result is (End-of-the-proof $\blacksquare$, established as desired. symbols) $\square$ ■ Q.E.A., ⊥ $Q. E. A.$, $\bot$ Multiplying both sides (Contradiction of the equation yields symbols) that 1 = 2. ⊥

6.2 Notational Symbols in Geometry and

Symbols LaTeX Code Example (Explanation)

◦ $^{\circ}$ cos(90◦) = 0 (Degree) !◦ 35 0 $'$ 350 = (Arcminute) 60 !0 20 00 $''$ 2000 = (Arcsecond) 60 rad $\mathrm{rad}$ π rad = 180◦ () grad $\mathrm{grad}$ 100 grad = 90◦ (Gradian)

6.3 Notational Symbols in Calculus

Symbols LaTeX Code Example (Explanation)

n2 + 1 +∞ $+\infty$ → +∞ (Positive infinity) n

6.3 Notational Symbols in Calculus 25

Comprehensive List of Mathematical Symbols

−∞ $-\infty$ lim ex = 0 x→−∞ (Negative infinity) ∆y ∆x $\Delta x$ m = (Change in ∆x variable) dx $\mathrm{d} x$ dy = f 0(x) dx (Differential) ∂f ∂x $\partial x$ dx (Partial ∂x differential) df $\mathrm{d} f$ dg(x, y) = (Total differential) ∂g ∂g dx + dy ∂x ∂y

6.4 Notational Symbols in Probability and Statistics

Symbols LaTeX Code Example (Explanation)

i.i.d. i.i.d. Given n i.i.d. random (Independent and variables X1,...,Xn, identically V (X1 + ··· + Xn) = distributed) V (X1) + ··· + V (Xn).

H0 $H_0$ H0 : µ = 23 (Null hypothesis) 2 6 2 Ha $H_a$ Ha : σ1 = σ2 (Alternative hypothesis)

7 Additional Resources

• Ultimate LaTeX Reference Guide: A definitive reference guide on the LaTeX language, with the commands, environments and

26 6.4 Notational Symbols in Probability and Statistics

Comprehensive List of Mathematical Symbols

packages most LaTeX users will ever need • Definitive Guide to Learning Higher Mathematics: A standalone 10-principle framework for tackling higher mathematical learning, thinking and problem solving • 10 Commandments of Higher Mathematical Learning: An illus- trated web guide on 10 scalable rules for learning higher mathe- matics • Definitive Glossary of Higher Mathematical Jargon: A tour around higher mathematics in 100 terms

6.4 Notational Symbols in Probability and Statistics 27

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